[不等式] 来自群的简单三元条件$x+y+z=xyz$求最小值

(14.74 KB)

2011-12-6 14:34

\begin{align*}
& x^{2}+y^{2}+z^{2}+\frac{2}{xyz}\\
=&\frac{(x^{2}+y^{2}+z^{2})(x+y+z)}{xyz}+2\sqrt{\frac{xyz}{(x+y+z)^{3}}} \\
\geqslant & \frac{(x+y+z)^{3}}{3xyz}+2\sqrt{\frac{xyz}{(x+y+z)^{3}}} \\
= & \left( \frac{1}{3}-\frac{1}{81\sqrt{3}} \right)\frac{(x+y+z)^{3}}{xyz}+\frac{1}{81\sqrt{3}}\cdot \frac{(x+y+z)^{3}}{xyz}+\sqrt{\frac{xyz}{(x+y+z)^{3}}}+\sqrt{\frac{xyz}{(x+y+z)^{3}}} \\
\geqslant & \left( \frac{1}{3}-\frac{1}{81\sqrt{3}} \right)\cdot 27+3\sqrt[3]{\frac{1}{81\sqrt{3}}} \\
= & 9+\frac{2\sqrt{3}}{9}.
\end{align*}